V. AISC 360-16 W Torsional Strength
Determine the torsional stress and maximum torsional rotation of a wide flange section per AISC Design Guide 9 using the LRFD method.
Details
A 15' long, simple-span beam is subject to a 15 kip load with a 6 inch eccentricity (e, with respect to the shear center) at mid-span. The section used is a W10x49 and the material is ASTM A992.
Validation
The eccentric load resolves into a 15 kip load through the shear center (Pu) combined with a torsional moment of . Assume that the section is pinned at the ends against torsion.
Section Properties
Taken from Table 1-1 of Ref. 1:
- Sx = 54.6 in3
- rts = 2.84 in
- ho = 9.42 in
- torsional constant, J = 1.39 in4
- warping constant, Cw = 2,070 in6
Taken from Appendix A of Ref. 2 (p.38):
- torsional constant, a = 62.1 in
- normalized warping function, Wno = 23.6 in2
- warping statical moment, Sw1 = 33.0 in4
- statical moment in flange (above web edge), Qf = 13.0 in3
- statical moment in web, Qw = 30.2 in3
Direct Bending Stress
Direct Shear Stress
In the web:
In the flanges:
Torsional Stresses
Evaluate the torsional function constants for ɑ = 0.5 for Case 3 using the charts in Appendix B of Ref. 2:
The shear stress due to pure torsion (St. Venant's), :
- At midspan,
- At the support,
- in the flange:
- in the web:
The shear stress due to warping,
- At midspan,
- At the support,
The normal stress due to warping,
- At midspan,
- At the support,
Combined Stresses
The maximum magnitude normal stress, fun, occurs at the midspan (where the normal stress is non-zero) within the flanges in which the bending and warping normal stresses are additive in magnitude:
The modification factor for σby and σw per Cl. 4.7.3 of Ref. 2 is given as:
where= | ||
= | ||
= | ||
= |
Thus, the modification factor is:
So the modified normal stress is taken as:
Check the limit state of yielding under normal stress per Cl. 4.7.1 of Ref. 2:
(4.12) |
The stress ratio is then:
The maximum magnitude shear stress, fuv, occurs at the support (where the torsion stress is non-zero):
- flanges:
- web:
Check the limit state of yielding under shear stress per Cl. 4.7.1 of Ref. 2:
(4.12) |
Calculate Twist
Occurs at midspan:
Results
Result Type | Reference | STAAD.Pro | Difference | Comments |
---|---|---|---|---|
Maximum normal stress, fun (ksi) | 40.4 | 40.855 | 1.1%a | |
Modified maximum normal stress, fun (ksi) | 43.8 | 44.280 | 1.1%a | |
Stress ratio for yielding under normal stress | 0.973 | 0.984 | 1.1%a | |
Stress ratio for yielding under shear stress | 0.422 | 0.416 | 1.4%a | |
Maximum rotation (radians) | 0.0937b | 0.099156 | 6%a | Difference in small numbers. |
- STAAD.Pro calculates the rotation and its derivatives using the formulas given in Appendix C of Design Guide 9, where as the calculations here as well as those used in the reference example use the charts provided in Appendix B.
- Ref. 2 calculates the twist at "service" level loads. Here, the twist calculation is made for the ultimate load.
STAAD.Pro Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2023\Samples \Verification Models\09 Steel Design\US\AISC\AISC 360-16 W Torsional Strength.STD is typically installed with the program.
The following design parameters are used:
- The value of SGR 28 specifies the use of ASTM A992 grade steel.
- The value of ALH 0.5 is used to indicate that the concentrated torque application location as a fraction of the beam length (here, mid-span, ɑ = 0.5).
- The value of CB 1.32 is used to specify a value of the Cb coefficient used in Chapter F in the AISC 360-16 code.
- The value of TORSION 1 is used to instruct STAAD.Pro to perform torsion checks.
- The value of TND 3 is used to indicate a torsion condition of concentrated torque with both ends pinned against torsion.
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 09-Sep-22
*Reference Design Guide 9 Example 5.1
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 15 0 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 4.176e+06
POISSON 0.3
DENSITY 0.489024
ALPHA 6.5e-06
DAMP 0.03
G 1.6128e+06
TYPE STEEL
STRENGTH RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
1 TABLE ST W10X49
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX MY MZ
LOAD 1 LOADTYPE None TITLE LOAD CASE 1
MEMBER LOAD
1 CON GY -2.5 7.5 0.5
LOAD 2 LOADTYPE None TITLE LOAD CASE 2
MEMBER LOAD
1 CON GY -7.5 7.5 0.5
LOAD COMB 3 COMBINATION LOAD CASE 3
1 1.2 2 1.6
PERFORM ANALYSIS
UNIT INCHES KIP
PARAMETER 1
CODE AISC UNIFIED 2016
ALH 0.5 ALL
CB 1.32 ALL
TORSION 1 ALL
SGR 28 ALL
TRACK 2 ALL
TND 3 ALL
CHECK CODE ALL
PRINT ANALYSIS RESULTS
FINISH
STAAD.Pro Output
STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2) ***************************************************** ALL UNITS ARE - KIP INCH (UNLESS OTHERWISE Noted). ***NOTE : AISC 360-16 Design Statement for STAAD.Pro. *** AXIS CONVENTION ***: ======================== The capacity results and intermediate results in the report follow the notations and axes labels as defined in the AISC 360-16 code. The analysis results are reported in STAAD.Pro axis convention and the AISC 360:16 design results are reported in AISC 360-16 code axis convention. AISC Spec. STAAD.Pro Description ------------ ----------- ------------- X Z Axis typically parallel to the sections principal major axis. Y Y Axis typically parallel to the sections principal minor axis. Z X Longitudinal axis perpendicular to the cross section. SECTION FORCES AXIS MAPPING: - AISC Spec. STAAD.Pro Description ------------ ----------- ------------- Pz FX Axial force. Vy FY Shear force along minor axis. Vx FZ Shear force along major axis. Tz MX Torsional moment. My MY Bending moment about minor axis. Mx MZ Bending moment about major axis. *** DESIGN MESSAGES ***: ======================= 1. Section classification reported is for the cross section and loadcase that produced the worst case design ratio for flexure/compression Capacity results. 2. Results for any Capacity/Check that is not relevant for a section/loadcase based on the code clause in AISC 360-16 will not be shown in the report. 3. Bending results are reported as being "about" the relevant axis (X/Y), while the results for shear are reported as being for shear forces "along" the axis. E.g : Mx indicates bending about the X axis, while Vx indicates shear along the X axis. *** ABBREVIATIONS ***: ====================== F-T-B = Flexural-Torsional Buckling L-T-B = Lateral-Torsional Buckling F-L-B = Flange Local Buckling W-L-B = Web Local Buckling L-L-B = Leg Local Buckling C-F-Y = Compression Flange Yielding T-F-Y = Tension Flange Yielding STAAD SPACE -- PAGE NO. 4 STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2) ***************************************************** ALL UNITS ARE - KIP INCH (UNLESS OTHERWISE Noted). - Member : 1 |-----------------------------------------------------------------------------| | Member No: 1 Profile: ST W10X49 (AISC SECTIONS)| | Status: PASS Ratio: 0.984 Loadcase: 3 | | Location: 90.00 Ref: DG9:Eq: 4.16(b) | | Pz: 0.000 T Vy: 7.500 Vx: 0.000 | | Tz: 0.1066E-14 My: 0.000 Mx: -675.0 | |-----------------------------------------------------------------------------| | TENSION SLENDERNESS | | Actual Slenderness Ratio : 70.677 | | Allowable Slenderness Ratio : 300.000 LOC : 0.00 | |-----------------------------------------------------------------------------| | STRENGTH CHECKS | | Critical L/C : 3 Ratio : 0.984(PASS) | | Loc : 90.00 Condition : DG9:Eq: 4.16(b | |-----------------------------------------------------------------------------| | SECTION PROPERTIES (LOC: 90.00, PROPERTIES UNIT: IN ) | | Ag : 1.440E+01 Axx : 1.120E+01 Ayy : 3.400E+00 | | Ixx : 2.720E+02 Iyy : 9.340E+01 J : 1.390E+00 | | Sxx+: 5.440E+01 Sxx-: 5.440E+01 Zxx : 6.040E+01 | | Syy+: 1.868E+01 Syy-: 1.868E+01 Zyy : 2.830E+01 | | Cw : 2.079E+03 x0 : 0.000E+00 y0 : 0.000E+00 | |-----------------------------------------------------------------------------| | MATERIAL PROPERTIES | | Fyld: 50.000 Fu: 65.000 | |-----------------------------------------------------------------------------| | Actual Member Length: 180.000 | | Design Parameters (Rolled) | | Kx: 1.00 Ky: 1.00 NSF: 1.00 SLF: 1.00 CSP: 12.00 | |-----------------------------------------------------------------------------| | COMPRESSION CLASSIFICATION (L/C: 1 LOC: 180.00) | | λ λp λr CASE | | Flange: NonSlender 8.93 N/A 13.49 Table.B4.1a.Case1 | | Web : NonSlender 23.18 N/A 35.88 Table.B4.1a.Case5 | | | | FLEXURE CLASSIFICATION (L/C: 1 LOC: 180.00) | | λ λp λr CASE | | Flange: Compact 8.93 9.15 24.08 Table.B4.1b.Case10 | | Web : Compact 23.18 90.55 137.27 Table.B4.1b.Case15 | |-----------------------------------------------------------------------------| STAAD SPACE -- PAGE NO. 5 STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2) ***************************************************** ALL UNITS ARE - KIP INCH (UNLESS OTHERWISE Noted). - Member : 1 Contd. |-----------------------------------------------------------------------------| | CHECKS FOR AXIAL TENSION | |-----------------------------------------------------------------------------| | TENSILE YIELDING | | DEMAND CAPACITY RATIO REFERENCE L/C LOC | | -0.000 648.0 0.000 Cl.D2 1 0.00 | | | | Intermediate Results : | | Nom. Ten. Yld Cap : Pn = 720.00 kip Eq.D2-1 | |-----------------------------------------------------------------------------| | TENSILE RUPTURE | | DEMAND CAPACITY RATIO REFERENCE L/C LOC | | -0.000 702.0 0.000 Cl.D2 1 0.00 | | | | Intermediate Results : | | Effective area : Ae = 14.400 in2 Eq.D3-1 | | Nom. Ten. Rpt Cap : Pn = 936.00 kip Eq.D2-2 | |-----------------------------------------------------------------------------| | CHECKS FOR AXIAL COMPRESSION | |-----------------------------------------------------------------------------| | FLEXURAL BUCKLING X | | DEMAND CAPACITY RATIO REFERENCE L/C LOC | | 0.000 571.6 0.000 Cl.E3 1 0.00 | | | | Intermediate Results : | | Effective Slenderness : Lcx/rx = 41.416 Cl.E2 | | Elastic Buckling Stress : Fex = 166.86 ksi Eq.E3-4 | | Crit. Buckling Stress : Fcrx = 44.106 ksi Eq.E3-2 | | Nom. Flexural Buckling : Pnx = 635.13 kip Eq.E3-1 | |-----------------------------------------------------------------------------| | FLEXURAL BUCKLING Y | | DEMAND CAPACITY RATIO REFERENCE L/C LOC | | 0.000 449.7 0.000 Cl.E3 1 0.00 | | | | Intermediate Results : | | Effective Slenderness : Lcy/ry = 70.677 Cl.E2 | | Elastic Buckling Stress : Fey = 57.298 ksi Eq.E3-4 | | Crit. Buckling Stress : Fcry = 34.701 ksi Eq.E3-2 | | Nom. Flexural Buckling : Pny = 499.70 kip Eq.E3-1 | |-----------------------------------------------------------------------------| | FLEXURAL-TORSIONAL-BUCKLING | | DEMAND CAPACITY RATIO REFERENCE L/C LOC | | 0.000 517.3 0.000 Cl.E4 1 0.00 | | | | Intermediate Results : | | Elastic F-T-B Stress : Fe = 92.875 ksi Eq.E4-2 | | Crit. F-T-B Stress : Fcr = 39.913 ksi Eq.E3-2 | | Nom. Flex-tor Buckling : Pn = 574.74 kip Eq.E4-1 | |-----------------------------------------------------------------------------| STAAD SPACE -- PAGE NO. 6 STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2) ***************************************************** ALL UNITS ARE - KIP INCH (UNLESS OTHERWISE Noted). - Member : 1 Contd. |-----------------------------------------------------------------------------| | CHECKS FOR SHEAR | |-----------------------------------------------------------------------------| | SHEAR ALONG X | | DEMAND CAPACITY RATIO REFERENCE L/C LOC | | 0.000 302.4 0.000 Cl.G1 1 0.00 | | | | Intermediate Results : | | Coefficient Cv Along X : Cv = 1.0000 Eq.G2-9 | | Coefficient Kv Along X : Kv = 1.2000 Cl.G6 | | Nom. Shear Along X : Vnx = 336.00 kip Eq.G6-1 | |-----------------------------------------------------------------------------| | SHEAR ALONG Y | | DEMAND CAPACITY RATIO REFERENCE L/C LOC | | 7.500 102.0 0.074 Cl.G1 3 0.00 | | | | Intermediate Results : | | Coefficient Cv Along Y : Cv = 1.0000 - | | Coefficient Kv Along Y : Kv = 5.3400 Eq.G2-5 | | Nom. Shear Along Y : Vny = 102.00 kip Eq.G2-1 | |-----------------------------------------------------------------------------| STAAD SPACE -- PAGE NO. 7 STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2) ***************************************************** ALL UNITS ARE - KIP INCH (UNLESS OTHERWISE Noted). - Member : 1 Contd. |-----------------------------------------------------------------------------| | CHECKS FOR BENDING | |-----------------------------------------------------------------------------| | FLEXURAL YIELDING (X) | | DEMAND CAPACITY RATIO REFERENCE L/C LOC | | -675.0 2718. 0.248 Cl.F2.1 3 90.00 | | | | Intermediate Results : | | Nom Flex Yielding Along X : Mnx = 3020.0 kip-in Eq.F2-1 | |-----------------------------------------------------------------------------| | FLEXURAL YIELDING (Y) | | DEMAND CAPACITY RATIO REFERENCE L/C LOC | | -0.000 1273. 0.000 Cl.F6.1 1 0.00 | | | | Intermediate Results : | | Nom Flex Yielding Along Y : Mny = 1415.0 kip-in Eq.F6-1 | |-----------------------------------------------------------------------------| | LAT TOR BUCK ABOUT X | | DEMAND CAPACITY RATIO REFERENCE L/C LOC | | -675.0 2718. 0.248 Cl.F2.2 3 90.00 | | | | Intermediate Results : | | Nom L-T-B Cap : Mnx = 3020.0 kip-in Eq.F2-2 | | Mom. Distr. factor : CbX = 1.3200 Custom | | Limiting Unbraced Length : LpX = 107.95 in Eq.F2-5 | | coefficient C : Cx = 1.0000 Eq.F2-8a | | Effective Rad. of Gyr. : Rts = 2.8462 in Eq.F2-7 | | Limiting Unbraced Length : LrX = 380.63 in Eq.F2-6 | |-----------------------------------------------------------------------------| STAAD SPACE -- PAGE NO. 8 STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2) ***************************************************** ALL UNITS ARE - KIP INCH (UNLESS OTHERWISE Noted). - Member : 1 Contd. |-----------------------------------------------------------------------------| | CHECKS FOR AXIAL BEND INTERACTION | |-----------------------------------------------------------------------------| | COMBINED FORCES CLAUSE H1 | | RATIO CRITERIA L/C LOC | | 0.248 Eq.H1-1b 3 90.00 | | | | Intermediate Results : | | Modification Fact. for Cb : Cbf = 1.0000 Eq.H1-2 | | Axial Capacity : Pc = 648.00 kip Cl.H1.1 | | Moment Capacity : Mcx = 2718.0 kip-in Cl.H1.1 | | Moment Capacity : Mcy = 1273.5 kip-in Cl.H1.1 | |-----------------------------------------------------------------------------| STAAD SPACE -- PAGE NO. 9 STAAD.PRO CODE CHECKING - AISC 360-16 LRFD (V1.2) ***************************************************** ALL UNITS ARE - KIP INCH (UNLESS OTHERWISE Noted). - Member : 1 Contd. |-----------------------------------------------------------------------------| | CHECKS FOR TORSION | |-----------------------------------------------------------------------------| | TORSION - NORMAL STRESS RATIO | | RATIO CRITERIA L/C LOC | | 0.984 DG9:Eq: 4.12 3 90.00 | | | | Intermediate Results : | | Torque : Tq = 45.000 kip-in DG9:Eq: 3.1 | | Twist : Twist = 0.99156E-01 DG-9: App. C | | Unfactored. Fcr : UFcr = 40.855 ksi DG-9: App. C | | Mod. Factor : Phi = 1.1204 DG9:Eq: 4.22 | | Factored. Fcr : Fcr = 44.280 ksi DG9:Eq: 4.8(a) | |-----------------------------------------------------------------------------| | TORSION - SHEAR STRESS RATIO | | RATIO CRITERIA L/C LOC | | 0.416 DG9:Eq: 4.13 3 180.00 | |-----------------------------------------------------------------------------| | TORSION - COMB. BUCK. RATIO | | RATIO CRITERIA L/C LOC | | 0.984 DG9:Eq: 4.16(b) 3 90.00 | |-----------------------------------------------------------------------------|